Monetary Policy and Stock Market Boomslchrist/research/...Monetary Policy and Stock Market Booms 87 According to the conventional wisdom, it is only the “excessive” booms that

85

Monetary Policy and Stock Market Booms

Lawrence Christiano, Cosmin Ilut, Roberto Motto and Massimo Rostagno

1. Introduction and Summary

The interaction between monetary policy and asset price vola-tility has been a matter of increased concern since the collapse of stock market booms in 2000 and 2007. Are booms like these sub-optimal? Is monetary policy partially responsible for stock market booms? Should monetary policy actively seek to stabilize stock mar-ket booms? These classic questions have been put back on the table by the experience of the past two decades.

1.1. The Conventional Wisdom

There is, we believe, a conventional wisdom on the answers to these questions. Booms arise for reasons largely unrelated to the conduct of monetary policy. Some booms are indeed excessive. But, it is unwise to identify which booms are excessive and to actively resist them us-ing interest rate policy. The conventional wisdom is that, in any case, a strategy of raising the policy interest rate when the inflation fore-cast is high and reducing it when the inflation forecast is low should help to dampen excessive volatility. The notion is that booms that are excessive involve a rise in stock prices above levels justified by funda-mentals. Such a boom represents a surge in demand because there is nothing currently on the supply side of the economy to justify it. In a

Abstract (not included in published document)

Historical data and model simulations support the following conclusion. Inflation is

low during stock market booms, so that an interest rate rule that is too narrowly focused

on inflation destabilizes asset markets and the broader economy. Adjustments to the

interest rate rule can remove this source of welfare-reducing instability. For example,

allowing an independent role for credit growth (beyond its role in constructing the

ináation forecast) would reduce the volatility of output and asset prices.

Published in 2010 Jackson Hole, "Macroeconomic Challenges: The Decade Ahead",

Federal Reserve Bank of Kansas City

86 Lawrence Christiano, Cosmin Ilut, Roberto Motto and Massimo Rostagno

demand boom, however, one expects inflation to be high. The policy of inflation forecast targeting using an interest rate rule “leans against the boom” at precisely the right time. This conventional wisdom was given an intellectually coherent foundation in two very influential papers (Bernanke and Gertler, 1999 and 2001).

1.2. Data

We explore an alternative perspective on the relationship between monetary policy and booms. We are motivated to consider this al-ternative by the historical record of U.S. stock market booms as well as by the Japanese stock market boom of the 1980s. We find that inflation was relatively low in each of the 18 U.S. stock market boom episodes that occurred in the past two centuries.1 The Japanese case is particularly striking, with inflation slowing sharply during the boom from its pre-boom level. The notion that stock market booms are not periods of high inflation, and that they are, if anything, periods of low inflation is not new to this paper. The recent work of Adalid and Detken (2007), Bordo and Wheelock (2004, 2007) and White (2009) also draws attention to this observation. Here, we stress the implications for monetary policy. The historical record suggests that, at least at an informal level, a monetary policy that implements infla-tion forecast-targeting using an interest rate rule would actually de-stabilize asset markets. The lower-than-average inflation of the boom would induce a fall in the interest rate and thus amplify the rise in stock prices in the boom.

A noticeable feature of stock market booms is that, with the excep-tion of only two of the 18 booms in our U.S. data set, credit growth is always stronger during a boom than outside a boom. On average, credit growth is twice as high in booms than it is in non-boom pe-riods. Casual reasoning suggests that volatility would be reduced if credit growth were tightened as booms get under way. In practice, this tightening in response to credit growth would not be justifiable based on the inflation outlook alone because booms are not in fact periods of elevated inflation. The idea that credit growth should be assigned an independent role in monetary policy has been advocated in several papers. We have advocated this position in work that we build on here (Christiano, Ilut, Motto and Rostagno, 2008).2

Monetary Policy and Stock Market Booms 87

According to the conventional wisdom, it is only the “excessive” booms that are inflationary. Assuming at least some of the booms considered in this paper are excessive, our results contradict the con-ventional wisdom that inflation accelerates during such booms. In-deed, the results raise the possibility that monetary policy is in part responsible for at least some booms by responding to the fall in infla-tion with interest rate cuts.

1.3. Interpreting the Data with a New Keynesian Model

Our empirical results raise an important question. How could it be that a stock market boom based purely on expectations about the future, which is therefore driven purely by demand, would not raise inflation? At first glance, the apparent finding that inflation is low during such booms may appear simply odd. Without a coherent framework to make sense out of it, one is reluctant to make an ap-parent anomaly the foundation for constructing a monetary policy strategy. This is why we turn to model simulations.

We show that the standard New Keynesian model provides an in-tellectual foundation for the notion that inflation is relatively weak in a boom. This is so, even in a boom that is based only on opti-mistic (possibly ill-founded) expectations about the future and not on real current developments. Our simulations provide support for the notion that a monetary policy that focuses heavily on inflation can exacerbate booms. No doubt there exist improvements to bank-ing supervision and credit market regulations that can moderate as-set price volatility.3 However, it seems inefficient to use supervision and regulation to remove volatility injected by monetary policy. That source of volatility could instead be removed by an adjustment to monetary policy.

We begin with the simplest possible New Keynesian model, the one analyzed in Clarida, Gali and Gertler (1999) and Woodford (2003). Because this model does not have capital in it, we cannot use it to think about a stock market boom. Still, we can use the model to think about booms driven by only optimism about the future, why inflation might be low at such a time, and how an inflation forecast-targeting interest rate rule might be destabilizing under these


circumstances.4 This analysis gets at the core of the issue—how infla-tion could be low in a demand-driven boom—and creates the ba-sic intuitive foundation for understanding the later results based on models that do incorporate asset prices.

We assume that people receive a signal that leads them to expect that a cost-saving technology will become available in the future. In the model, prices are set as a function of current marginal costs as well as future marginal costs. The expectation that marginal costs in the future will be lower dampens the current rise in prices. The infla-tion forecast-targeting interest rate rule leads the monetary authority to cut the interest rate, stimulating the demand for goods. Output expands to meet the additional demand, raising current marginal costs. The expected future reduction in marginal cost exceeds the current rise, so that prices actually fall during the boom.5

That prices are set in part as a function of future marginal costs is essential to our analysis. In the model, forward-looking price set-ting reflects the presence of price adjustment frictions. However, it is easy to think of other reasons why price-setters might be forward looking. For example, firms may be motivated to seek greater market share in order to be in a better position in the future to profit from anticipated new future technologies. The drive for greater market share may lead to a pattern of price cutting. This particular strategy for responding to anticipations of improved technology is one that has, for example, been stressed by Jeff Bezos, the CEO of Amazon.6

1.4. Why an Inflation Forecast-Targeting Interest Rate Rule May Destabilize a Boom

The boom that occurs in the wake of a signal about future tech-nology in our model simulation is excessive in a social welfare sense. Its magnitude reflects the suboptimality of the inflation forecast-tar-geting interest rate rule. The monetary policy that maximizes social welfare responds to the optimistic expectations by raising the real interest rate sharply (we refer to the socially optimal interest rate as “the natural rate of interest”). The reason for the sharp rise in the natural rate of interest is simple. The expectation of higher fu-ture consumption opportunities creates the temptation to increase


consumption right away. But, such an increase is inefficient be-cause the basis for it—improved technology—is not yet in place. In a world where markets operate smoothly, the efficient outcome—a delay in the urge to consume—is automatically brought about by a rise in the real interest rate. In such a world, the natural and actual rates of interest coincide. In the world of our model, the smooth op-eration of markets is hampered by price and wage frictions, and the monetary authority’s control over the nominal rate of interest gives it control over the real interest rate. This control can be used for good or ill: The monetary authority has the power to make the real rate of interest close to or far from the natural rate of interest. The mon-etary authority using an inflation forecast-targeting interest rate rule responds to the signal about future productivity in exactly the wrong way. The monetary authority observes downward pressure on infla-tion in the wake of the signal and responds by reducing the interest rate. The difference between the high interest rate that is optimal and the low interest rate that actually occurs represents a substantial and socially suboptimal monetary stimulus. The boom that occurs in the wake of a signal of improved future technology is largely a phenom-enon of loose monetary policy in our model.

One way to characterize the problem with the inflation forecast- targeting interest rate rule is that the rule does not assign any weight to the natural rate of interest or to any variable that is well correlated with it. Traditionally, the absence of the natural rate of interest from interest rate rules is motivated on two grounds. First, in practice, this variable is hard to measure because it depends on hard-to-determine details about the structure of the economy. Second, in much of the model analysis that appears in the existing literature, the natural rate of interest fluctuates relatively little and so approximating it by a constant does not represent a very severe mistake. Regarding the first consideration, we argue that credit growth may be a good proxy for the natural rate.

Consider the second motivation for ignoring the natural rate of interest in an interest rate rule. Until recently, builders of models have assumed that shocks to the demographic factors that influence labor supply, to government spending and to the technology for


producing goods and services occur without advance warning. We confirm that the natural rate of interest fluctuates relatively little in response to shocks that occur without warning. However, recently there has been increased attention to the possibility that people receive advance signals about shocks.7 Consider the case of shocks to government spending and to technology. The major government spending shocks are associated with wars. When that kind of spend-ing jumps—the troops are on the move and the bullets are flying—it does so after a lengthy period of increased tensions and political ma-neuvering. These events prior to actual increases in war spending rep-resent the early signals about government spending.8 Disturbances in technology work in the same way. Signals that the information technology revolution would transform virtually everything about how business is done existed decades ago.9 We show below that the natural rate of interest fluctuates a lot more in response to a signal about a future shock than it does to a shock that occurs without any advance warning.10 That is, when we take seriously that many dis-turbances occur with advance warning, the assumption of a constant natural rate of interest in an interest rate rule is no longer tenable.

So, the problem with the inflation forecast-targeting interest rate rule is that it reduces the interest rate in a boom triggered by optimis-tic expectations, while the efficient monetary policy would increase the interest rate.11 Paradoxically, we first develop this finding below in a model with only price frictions, in which the optimal monetary policy (i.e., the policy that sets the interest rate equal to the natural rate) completely stabilizes inflation. That is, our analysis does not necessarily challenge the wisdom of inflation per se, only the effec-tiveness of doing so with an inflation forecast-targeting interest rate rule that is principally driven by the inflation forecast.12

1.5. Why Adding Credit Growth to the Interest Rate Rule May Help

Up to this point, the analysis has focused on models that are suffi-ciently simple that they can be analyzed with pen and paper. We then verify the robustness of the analysis by redoing it in a medium-sized New Keynesian dynamic stochastic general equilibrium (DSGE) model that incorporates capital and various frictions necessary for it


to fit business cycle data well. In this model, optimism about the fu-ture triggers a fall in inflation and a rise in output, the stock market, consumption, investment and employment. The boom is primarily an artifact of the empirically estimated interest rate policy rule, in which the forecast of inflation is assigned an important role. Under the optimal monetary policy, the boom would involve only a modest rise in output, and this would be accomplished by a sharp rise in the rate of interest.

We use the medium-sized model to investigate the possibility, sug-gested by the historical data record, that assigning a role —beyond its role in forecasting inflation —to credit growth may help to stabilize booms. First, however, we must modify the model to incorporate an economically interesting role for credit. We do so by introducing financial frictions along the lines suggested in the celebrated contri-bution by Bernanke, Gertler and Gilchrist (1999) (BGG). We obtain the same results in this model that we found in our simple model and in the model with capital. The inflation forecast-targeting interest rate rule causes the economy to overreact to the optimism about the future, though inflation during the boom is low. The natural rate of interest rises sharply in the model. When we assign a separate role for credit growth in the interest rate rule, then the response of the economy is more nearly optimal. We interpret this as signifying that credit growth is a reasonable proxy for the natural rate of interest.

1.6. Organization of the Paper

The paper is organized as follows. The first section below describes the data. The following section describes the analysis of our simple model. Our analysis features a baseline parameterization, but also examines the robustness of the argument to perturbations. We con-sider, for example, interest rate rules that look at inflation forecasts as well as at current inflation. We also consider the case where price stickiness arises because of frictions in the setting of wages rather than because of frictions in price setting per se. This is an important perturbation to consider because empirical analyses typically find that it is crucial to include wage stickiness if one is to fit the data well. The next section considers the analysis of the expanded model


with credit and asset markets. We offer concluding remarks at the end. Technical details are relegated to the Appendix.

2. Inflation and Credit Growth in Stock Market Booms: The Evidence

This section displays data on stock market boom-bust episodes. We find that, in all cases, inflation is relatively low during the boom phase in these episodes. Real credit growth was relatively high in all but two episodes. We also examine data on the Japanese stock market boom in the 1980s. As in all U.S. stock market booms, this Japanese boom is associated with a drop in inflation. Presumably, the boom was fueled in part by the accommodative Japanese monetary policy of the time, which cut short-term interest rates substantially. We show that if the Bank of Japan had followed a standard interest rate rule that assigns weight to inflation and also the output gap, then its interest rate would have been cut even more sharply. The Japanese experience of the 1980s presents perhaps the most compelling em-pirical case for the proposition that an interest rate rule that focuses on the forecast of inflation exacerbates stock market volatility.13

We split our U.S. data set into two parts. The first part covers 12 episodes in the 19th and early 20th centuries, and the second consid-ers four episodes beginning with the Great Depression. We divide our data set in this way because we have annual observations for the first part and quarterly observations for the second part. In addition, data availability considerations requires that our concepts of credit differ slightly between the two periods.

Consider the first part of our data, which is displayed in Chart 1. The stock market index is the log of Schwert’s (1990) index of com-mon stock, after deflating by the consumer price index.14 The real output measure is the logarithm of real gross national product.15 Our measure of real credit is the quantity of bank loans, scaled by the con-sumer price index.16 We define a stock market boom-bust episode as follows. We start with 12 financial panics in the 19th century and the pre-World War I portion of the 20th century.17 These are indicated by a solid circle in Chart 1, and they are listed in Table 2. Although each panic is associated with a drop in the stock market, Chart 1


indicates that in all but three cases, the stock market had already be-gun to drop before. We define the peak associated with a particular financial panic as the year before the panic when the stock market reached a local maximum. We define the trough before the peak as the year when the stock market reached a local minimum. The pe-riod bracketed by the trough and the peak associated with a financial panic is indicated in Chart 1 by a shaded area. In addition, we block from our analysis the period of the Civil War, which is indicated by its own shaded area.

We can see from Chart 1 that in virtually every stock market boom, the price level actually declined. Moreover, in no case did the price level rise more than its average in the non-boom, non-Civil War pe-riods. In addition, we see that stock market booms are typical periods of accelerated credit growth. Table 1 quantifies the findings in Chart 1. According to that table, consumer price index (CPI) inflation aver-aged -2.5 percent during stock market booms, substantially less than the 0.7 percent inflation that occurred on average over non-boom periods. In addition, credit grew twice as fast, on average, during a stock market boom as during other periods. Table 1 shows just how volatile the stock market was over this period. It grew at a 10-percent

Chart 1Data for 19th and Early 20th Century

Notes: (1) Shaded areas correspond to stock market booms. (2) A constant has been added to each variable (after logging) to spread out the time series in the graph and improve visibility. (3) Civil War is indicated in the figure and excluded from calculations


Table 1Variables Over Various Subperiods, 1803-1914

Table 2Variables in Stock Market Boom Episodes

Periods CPI Credit GDP Stock Price

Boom -2.5 9.5 4.6 10.2

Other 0.7 4.0 3.1 -6.3

Non-Civil War -0.7 6.5 3.7 0.8

Notes: (1) Numbers represent 100 times average of log first difference of indicated variable over indi-cated period. (2) Boom periods are the union of the trough-to-peak periods enumerated in Table 2.(3) “Other” are periods that are not booms and that fall outside of 1861-1865. (4) Results for credit data based on period 1819-1914 due to data availability.

A. Non-Boom, Non-Civil War, 1803-1914

CPI Credit GDP Stock Price

0.7 4.0 3.1 -6.3

B. Boom Episodes

Panic Trough-Peak CPI Credit GDP Stock Price

1819 1814-1818 -8.0 na 1.8 9.8

1825 1822-1824 -9.8 21.9 3.7 12.1

1837 1827-1835 -1.5 14.6 4.9 5.2

1857 1847-1852 -1.3 7.6 5.4 6.9

1873 1865-1872 -4.1 11.9 4.8 8.5

1884 1877-1881 -0.6 3.5 7.5 16.0

1890 1884-1886 -2.2 4.9 5.9 15.2

1893 1890-1892 0.0 5.6 4.5 7.9

1896 1893-1895 -3.3 4.2 4.4 3.9

1903 1896-1902 0.3 8.6 5.3 11.1

1907 1903-1905 0.0 7.6 2.3 18.3

1910 1907-1909 -1.8 4.0 0.6 25.1

Notes: (1) Numbers represent 100 times average of log first difference of indicated variable over indicated period. (2) Panel A: data averaged over period 1802-1914, skipping 1861-1865 years and trough-to-peak years. (3) Panel B: data averaged only over the indicated trough-peak years. (4) Panics occur after stock market peak. (5) NA signifies “not available,” observations on credit begin in 1819.


pace during boom periods and shrank at a 6.3-percent rate in non-booms. Table 2 provides a breakdown of the data across individual boom periods. The table documents the fact, evident in Chart 1, that there is little variation in the general pattern. Inflation is lower in every stock market boom than its average value outside of booms. In the case of credit, there is only one episode in which credit growth was slower in a stock market boom than its average outside of booms. That is the boom associated with the 1884 panic.

We now turn to the data for the post-World War I period.18 The data are displayed in Chart 2. We exclude the World War II period from our analysis, and this period is indicated by the shaded area. The other shaded areas indicate six stock market booms in the 20th and early 21st century. As in the earlier data set, each boom episode is a time of non-accelerating inflation. In several cases, inflation actu-ally slowed noticeably from the earlier period. Note, too, that stock market booms are a time of a noticeable increase in the growth rate of credit. These results in Chart 2 are quantified in Tables 3 and 4. According to Table 3, CPI inflation in stock market booms is half its value in other (non-World War II) times. Credit growth, as in the 19th century, is twice as rapid in boom times as in other times. According to the results in Table 4, inflation in each of the six boom episodes considered is below its average in non-boom times. With one exception, credit growth is at least twice as fast in booms as in other periods. The exception was the boom that peaked in 1937. This started in the trough of the Great Depression.

Chart 3 displays a real index of Japanese stock prices, as well as the Japanese CPI.19 The trough and peak of the 1980s boom corresponds to 1982Q3 and 1989Q4, respectively. The time of the boom is high-lighted in both Charts 3 and 4. What is notable about Chart 3 is that CPI inflation is significantly positive before the start of the 1980s stock market boom, and it then slows significantly as the boom pro-ceeds. Inflation even falls below zero a few times in the second half of the 1980s. We ask what a monetary authority that follows a standard inflation-targeting interest rate rule would have done in the 1980s. In particular, we posit the following policy rule for setting Japanese call money rate, Rt:


Chart 2Data for 20th and Early 21st Century

Table 3Variables Over Various Subperiods, 1919Q1-2010Q1

Periods CPI Credit GNP Stock Price

Boom 1.8 5.3 4.6 13.8

Other 4.0 2.3 0.2 -11.7

Whole period 2.7 4.0 2.7 2.7

Notes: (1) Numbers represent 100 times average of log first difference of indicated variable over in-dicated period. (2) Boom periods are the union of the trough-to-peak periods enumerated in Table 4. (3) “Other” are periods that are not booms and that exclude World War II (1939Q4-1945Q4). “Whole period” corresponds to the full sample, excluding World War II.

Notes: (1) Shaded areas correspond to stock market booms. (2) A constant has been added to each variable (after logging) to spread out the time series in the graph and improve visibility. (3) World War II is indicated in the chart and excluded from calculations.


Chart 3Japanese Stock Market Boom in the 1980s

Table 4Variables in Stock Market Boom Episodes

A. Non-Boom, Non-World War II, 1919Q1-2010Q1

CPI Credit GNP Stock Price

4.0 2.3 0.2 -11.7

B. Boom Episodes

Trough-Peak CPI Credit GNP Stock Price

1921Q3-1929Q3 -0.2 5.7 5.9 19.3

1932Q2-1937Q2 0.6 -2.1 6.5 24.2

1949Q2-1968Q2 2.0 6.3 4.2 8.1

1982Q3-1987Q3 3.2 7.5 4.3 17.5

1994Q2-2000Q2 2.5 6.1 3.9 16.4

2003Q1-2007Q1 3.0 4.6 3.0 10.1

Notes: (1) Numbers represent 100 times average of log first difference of indicated variable over in-dicated period. (2) Panel A: data averaged over period 1919Q1-2010Q1, skipping 1939Q4-1945Q4 and trough-to-peak years. (3) Panel B: data averaged only over the indicated trough-peak years. (4) Panics occur after stock market peak.


Rt = 0.7Rt-1 + (1 – 0.7) [R + 1.5 (Ut–U) + 0.5gapt ], (2.1)

where t denotes quarters; gapt denotes the output gap; and Ut denotes the actual, year-over-year rate of inflation. For R and U� we used the sample average of the call money rate and the inflation rate in the pe-riod immediately preceding the boom, 1979Q1-1982Q3. Also, we used the gap estimates produced by the International Monetary Fund in the process of preparing the World Economic Outlook.20 The re-sults are displayed in Chart 4.21 The starred line displays the actual call money rate, while the solid line displays the values of Rt that solve (2.1) over the period 1979Q1-1989Q4. Note that the Bank of Japan loosened policy very significantly during the boom, bringing the interest rate down on the order of 300 basis points. That action by the Bank of Japan is thought by many to have been a mistake, and to have contributed to a stock market boom that in retrospect ap-pears to have definitely been “excessive” (see, e.g., Shirakawa, 2010). But, note that if the Bank of Japan had implemented the policy rule, (2.1), they would have reduced the interest rate an additional 200 basis points over what they actually did do. One has to suppose that this would only have further destabilized an already volatile market. We hasten to add a caveat because we are conjecturing what would have happened under the counterfactual monetary policy rule, (2.1). Such a counterfactual experiment would have a host of general equi-librium consequences that might have changed the realized data in profound ways. This is why we now leave the informal analysis of data and turn to the analysis of models next.

3. A Simple Model For Interpreting the Evidence

We begin our analysis in a model that is simple enough that the core results can be obtained analytically, without the distraction of all the frictions required to fit aggregate data well. The model is a version of the workhorse model used in Clarida, Gali and Gertler (1999) (CGG) and Woodford (2003).

We posit that the driving disturbance is a “news shock,” a distur-bance to information about the next period’s innovation in technol-ogy.22 News that technology will improve in the future creates the ex-pectation that future inflation will be low, and this leads an inflation


forecast-targeting monetary authority to reduce the nominal rate of interest. This policy creates an immediate expansion in the economy. Although the expansion is associated with higher current marginal cost, inflation nevertheless drops in response to the lower future ex-pected marginal costs.

We obtain our results in this section under two specifications for why there are frictions in prices. In one scenario (“pure sticky pric-es”), there are frictions directly in the setting of prices. In this sce-nario, wages are set flexibly in a competitive labor market. In the second scenario (“pure sticky wages”), prices are set flexibly, but are influenced by frictions in the setting of wages. Our model of wage frictions is the one proposed in Erceg, Henderson and Levin (2000) (EHL). The inefficient boom with low inflation occurs in both sce-narios, though it does so across a wider range of parameter values under sticky wages.

The action of the monetary authority in reducing the nominal rate of interest in response to a news shock is exactly the wrong one in this model. Under the efficient monetary policy, the nominal rate of in-terest should not be decreased. Indeed, under pure sticky prices, the

Chart 4Japan, Actual Rate and Rate Implied by

Simple Interest Rate Rule


nominal rate of interest should be increased substantially in response to a news shock. In the model, it is efficient for employment to be constant in each period and for consumption to track the current realization of technology. The news shock triggers an expectation of higher future consumption, and the efficient rate of interest rises in order to offset the intertemporal substitution effects associated with an expectation of higher future consumption.

Household preferences in the model are:

E CL

tl

t lt l

Ll

L

βσ

σ

log( ) ,+++

=

∞

−+

⎡

⎣⎢

⎤

⎦⎥∑

1

10

where Ct denotes consumption and Lt denotes employment. The household budget constraint is:

PtCt + Bt+1 fWtLt + Rt-1Bt + Tt ,

where Tt denotes lump sum income from profits and government transfers, Rt denotes the nominal rate of interest and Pt ,Wt denote the price level and wage rate, respectively.

Final goods, Yt , are produced as a linear homogeneous function of Yit , i �(0,1) using the following Dixit-Stiglitz aggregator:

Y =t Y dlltf

f

0

11

∫⎡

⎣⎢⎢

⎤

⎦⎥⎥

λ

λ

.

A representative, competitive final good producer buys the ith inter-mediate input at price, Pit . The ith input is produced by a monopolist, with production function

Yit = exp (at )Lit.

Here, Lit denotes labor employed by the ith intermediate good pro-ducer. The ith producer is committed to sell whatever demand there is from the final good producers at the producer’s price, Pit. The pro-ducer receives a tax subsidy on wages in the amount, (1–v)Wt , where v is set to extinguish the monopoly distortion in steady state. The subsidy is financed by lump sum taxes on households.


In the pure sticky price version of the model, wages are set flex-ibly in competitive markets, and prices are set by the intermediate good monopolists, subject to Calvo-style frictions. In particular, with probability ]p the ith producer, i �(0,1), must keep its price unchanged to its value in the previous period, and with the comple-mentary probability, the producer can set its price optimally. In the pure sticky wage version of the model, intermediate good producers set prices flexibly, as a fixed markup over marginal cost. Following EHL, we adopt a slight change in the specification of household util-ity in which households are monopolists in the supply differentiated labor services indexed by j, j �(0,1), and they set wages subject to Calvo-style frictions. With probability ]w the wage of the jth type of specialized labor service cannot be changed from its value in the previous period. With the complementary probability, the wage rate of the jth specialized labor service is set optimally.

We consider these two extreme specifications of price/wage setting frictions because their simplicity allows us to derive results analytical-ly. We consider the case with both sticky wages and prices, as well as other features useful for fitting aggregate data well, in the next section.

In our baseline analysis, we adopt the following law of motion for at:

a a u ut t t t t t= + ≡ +− −ρ ξ ξ1 , .01

1

(3.1)

Here, ut represents the white noise one-step-ahead error in forecast-ing at based on its own past. We posit that this error is the sum of two mean-zero, white noise terms, ]t

0 and ξt −11 , where

E a E a st t s t t sξ ξ0− − −= = >1

1 0 0, .

The subscript on ξt1indicates the date when this variable is revealed

to agents in the model, j = 0, 1. Thus, at time t agents become aware of ]t

oand ]t

1 . Here, ]torepresents the last piece of information re-

ceived by agents about ut and ]t1 represents the first piece of informa-

tion about ut+1. We refer to ]t1 as “news.”

As is now standard, we express the household’s log-linearized inter-temporal Euler equation in deviation from what it is in the first-best


equilibrium—in which the inflation rate is always zero and the inter-est rate isRt

* � , as follows:

ˆ ˆ ˆ ˆ .*x E R R E xt t t t t t t= − − −⎡⎣

⎤⎦++ +π 1 1

(3.2)

Here, x t denotes the output gap, the percent deviation between the actual and efficient levels of output. Also, Rt

and Ut denote the per-cent deviation of the gross nominal interest rate and of the gross inflation rate, respectively, from their values in steady state. Similarly, Rt

* denotes the percent deviation of the gross nominal interest rate in the efficient equilibrium from its steady state.

As noted above, employment is constant in the efficient equilib-rium and consumption is proportional to exp (at ). In addition, infla-tion is zero. These properties, together with the assumption of unit intertemporal elasticity of substitution imply that, after linearization, Rt

* corresponds to the expected change in at :23

R E a a at t t t t t* .= − = −( ) ++1 ρ ξ1 1 (3.3)

The shock to current productivity, ]t0 , enters via at with a coefficient

of W–1. In standard empirical applications which do not incorporate news shocks, the values of autoregressive coefficients like W are es-timated to be large (in a neighborhood of 0.9), and as a result, Rt

*

is not very volatile. At the same time, note how the signal shock,]t

1 , appears with a unit coefficient in Rt* . Evidently, the introduc-

tion of news shocks may increase the volatility of Rt* by an order of

magnitude. The intuition is simple. A persistent shock that arrives without advance warning creates little incentive for intertemporal substitution. Such a shock creates only a small need to change the interest rate. By contrast, a signal that a persistent shock will occur in the future creates a strong intertemporal substitution motive, which requires a correspondingly strong interest rate response.

The simplest representation of an interest rate rule that focuses on inflation is the following:

ˆ ˆ .R a Et t t= +π π 1 (3.4)

This specification of the monetary policy rule, together with a par-ticular labor market subsidy explained in Appendix A, are structured so that the steady state of the efficient and actual equilibria coincide.


Completing the model requires an additional equation, a Phillips curve. We discuss the Phillips curve corresponding to pure sticky prices and pure sticky wages, respectively, in the following two sec-tions. The derivation of the equilibrium conditions is tedious, but well known. For completeness, we include them in the Appendix.

3.1. Pure Sticky Prices

The equilibrium condition associated with price-setting is, after linearization:

ˆ ˆ ˆ .π γ β πt t t tx E= + +1 (3.5)

The slope of the Phillips curve with respect to the output gap, L, is related to structural parameters as follows:

γξ βξ

ξσ=

−( ) −( ) +( )1 p p

pL

11 , (3.6)

where ] p is the probability that a firm cannot change its price. Also, 1 + XL represents the elasticity of firm marginal cost with respect to the output gap.

The price Phillips curve, (3.5), and IS relation, (3.2), after sub-stituting out for Rt and Rt

* , represent two equations in two un-knowns, Ut and x t . We posit the following solution,

π η φ ξπ πt t ta= + 1 (3.7)

ˆ ,x at x t x t= +η φ ξ1 (3.8)

where MU�, KU�,MH, KH are undetermined coefficients. The Appendix uses straightforward, though tedious, algebra to solve for these ob-jects. In the case of the response to at (hence, ]t

0 ):

ηψ ρβ ρ

γη γ

ρβηπx x= −

−( ) −( ) =−

1 1

1, .


Here,

ψ γρβ ρ γρπ

=−( ) −( )+ −( ) >

1 1 10

a.

It is evident that:

Proposition 3.1. Mx, MU < 0 for all admissible parameter values.

Simple substitution implies the following solution for the interest rate:

ˆ ,R a a at t t= −( ) + −( )π πψ ρ ρ ψ ρ ξ1 1 1 (3.9)

It is interesting to compare the actual interest rate response, Rt in (3.9), with the efficient interest rate response, Rt

* in (3.3). We can see that if aU is sufficiently large, then aU^(W–1) W�q�W��1 and the interest

rate response to at (and, hence, to ]t0) is efficient. For more moder-

ate values of aU� the interest rate at least has the right sign response to at , though the magnitude of that response is inefficiently weak. By

contrast, the response of Rt to ]t1is perverse. As noted above, the effi-

cient interest rate displays a strong and positive response to ]t1 , while

Rt remains unchanged for W = 1 and actually declines for W < 1. To understand the perverse response of the interest rate to a news shock, we need to first discuss the reduced form parameters, Mx ,MU��KU��Kx .

Consider Mx ,MU. Proposition 3.1 implies that ]todrives both the out-

put gap and inflation down. The intuition for this result is straightfor-ward. Given the assumed time series representation for at , a positive shock to ]t

o raises at and creates the expectation that at will be smaller in later periods. Relative to the efficient intertemporal consumption path in which ct = at, households wish to reallocate consumption into the future. The monetary policy rule offsets the relative weakness in period t demand by reducing the interest rate, Rt, but the response is not strong enough. As a result, period t spending expands by less than the rise in at, accounting for the fall in the output gap in period t. The fall in the output gap implies weak labor demand and, hence, low labor costs. The reduction in costs accounts for the drop in inflation.


The fact, MU < 0, explains why Rt drops in response to ]t1 . The

news shock creates the expectation that technology will be launched on a temporary high in the next period, creating the expectation that inflation in the next period will be low. This is evident by evaluating (3.7) in t + 1 and taking the period t conditional expectation:

E at t t tˆ .π η ρ ξπ+ = +( )1

1

Because a positive disturbance to ξt1

reduces anticipated inflation, and because our assumed monetary policy rule reacts to the inflation forecast, it follows that Rt drops in response to a positive innovation in ]t

1 .

The remaining reduced form parameters, KU�and Kx, control the re-sponse of inflation and the output gap to a news shock, ]t

1 . Appendix A establishes that these parameters are given by:

φ ψ φ ψ β ρ γπ π πx a a= −( ) = − −( )+ −( )⎡⎣ ⎤⎦1 1, .1 (3.10)

From the first expression, we see that Kx> 0, so that the output gap always jumps with a positive signal about future productivity, ]t

1 . The consumption-smoothing motive and the rise in expected future consumption create a desire to increase current spending. In the ef-ficient equilibrium the interest rate, Rt

* , increases sharply in order to keep spending equal to the unchanged current value of at. But, as discussed earlier, Rt either does not respond at all in the limiting case, W = 1; or it actually falls.

Turning to KU��the impact of ]t1 on Ut operates by way of its effects

on current and future marginal cost. These effects can best be seen by solving the Phillips curve forward and making use of (3.8) and the law of motion for at :

24

ˆ ˆ ˆ ˆ ˆπ γ β β β

γηπ

t t t t t t t tx E x E x E x= + + + +⎡⎣ ⎤⎦

=

+ + +12

23

3 #

111

1 1

1−+ +

−βργφ ξ γ βη

βρξπat x t t (3.11)

The first term involving ]t1 , LKx,pertains to the impact of a news

shock on date t marginal cost. This term is definitely positive because


a positive period t news shock raises the period t output gap (recall, Kx > 0). Thus, the impact of the news shock on Ut is positive if we only take into account period t marginal cost (i.e., if G = 0). Note that the second term involving ]t

1 is definitely negative (recall, MU < 0). This term reflects that a positive realization of ]t

1signals a fall in

future marginal costs. Thus, the net effect on current inflation of ]t1

is ambiguous, and so we must turn to a numerical example.

The intuition sketched in this section suggests that the sign of the period t inflation and output response to ]t

1 is likely to be sensitive to the assumptions about the time series representation of at. Suppose, for example, that at+1 > at after a positive shock to ]t

o In this case, the shock to ]t

0 is likely to trigger a surge in the demand for goods, mak-ing Mx and MU positive.25 This, in turn, suggests that in the period of a jump in ]t

1, firms would anticipate a rise in marginal cost not only in the current period but in future periods as well, so that Ut would increase. We explore the robustness of our results to the assumptions about at in the numerical experiments below.

3.2. Pure Sticky Wages

We now consider the case in which prices are flexible, but there are frictions in the setting of wages, as spelled out in EHL. They derive the following equilibrium condition:

ˆ ( ) ˆ,π

ξ βξ

ξ σ λλ

σw tw w

w Lw

w

L tx=−( ) −( )+

−⎛

⎝⎜

⎞

⎠⎟

+ −1 1

11

1 ˆ ˆ ,,wt w t⎡⎣ ⎤⎦+ +βπ 1

(3.12)

where Uw,t denotes the gross growth rate of the nominal wage rate and wt denotes the real wage, divided by technology, exp (at). As before, a hat over a variable indicates percent deviation from steady state. For completeness, (3.12) is derived in the Appendix. The intuition for (3.12) is straightforward. The first object in the square brackets is the real marginal cost of work scaled by the technology shock, expressed in percent deviation from steady state.26 It is perhaps not surprising that when this object is higher than the scaled real wage, nominal wage growth is high. The growth rate of the scaled real wagewt , the


price level, the nominal wage rate and the state of technology are related by the following identity:

ˆ ˆ ˆ ˆ,w w a at t w t t t t= + − − −( )− −1 π π 1

(3.13)

With flexible prices, (3.5) drops from the system. In addition, the fact that price-setters set prices as a fixed markup over marginal cost implies wt =0 for all t. Imposing this condition and rearranging, we find, using (3.3):

E E Rt w t t t tˆ ˆ .,

*π π+ += +1 1 (3.14)

Rewriting (3.12) taking wt " 0 into account, we obtain:

ˆ ˆ ˆ ,, ,π γ βπw t w t w tx= + +1 (3.15)

where

γξ βξ

ξ σ λλ

σww w

w Lw

w

L=−( ) −( )+

−⎛

⎝⎜

⎞

⎠⎟

+( )1 1

11

1 .

(3.16)

We see an important distinction here between sticky wages and sticky prices. For a given degree of stickiness in wages and prices, i.e., ]p = ]w, the slope of the wage Phillips curve, (3.15), is smaller than the slope of the price Phillips curve, (3.5). The intuition for this is simple. Because of constant returns to scale, firms in this economy have constant marginal costs. The marginal cost of supplying labor, by contrast, is increasing in labor and is steeper for larger XL. The price set by a monopolist with a steep marginal cost curve reacts less to a cost shock than does the price set by a monopolist with flat marginal cost. This effect on the monopolist’s price response is mag-nified when demand is highly elastic and explains the presence of the elasticity of demand for labor in (3.16), Qw/ (Qw–1).27,28

Using (3.14) to replace price inflation with wage inflation in the policy rule and the IS equation (see (3.4) and (3.2)),

ˆ ˆ,

*R a E Rt t w t t= −⎡⎣ ⎤⎦+π π 1 (3.17)


ˆ ˆ ˆ ˆ .,x E R E xt t t w t t t= − −( )++ +π 1 1 (3.18)

The three equilibrium conditions associated with the pure sticky wage model are the wage Phillips curve, (3.15); the policy rule, (3.17); and the IS equation, (3.18). This system can be solved for ˆ , ˆ ˆ

,x Rt w t tandU . The implications for price inflation can then be de-duced using (3.13) and wt " 0.

The solution of the system can be represented as follows:

ˆ , ˆ ,,π η φ ξ η φ ξπ πw tw

tw

t t xw

t xw

ta x a= + = +1 1 (3.19)

as in (3.10), with

ˆ ˆ ,,π πt w t t ta a= − −( )−1 (3.20)

according to (3.13). By this last expression, the impact of ]t1 on Ut is

simplyφπw. Section A2 in the Appendix establishes:

η γβρ

η ηρ βρ

ρ βρππ

π

w wxw

xw a

a=

−=

− −( ) −( )−( ) −( )+ −1

1 1

1 1,

111 1( ) = − −( ) −( )ργ

ψγ

ρ βρπw

w

w

a .

Evidently, the analog of Proposition 3.1 holds for sticky wages:

Proposition 3.2. η ηπw

xw, < 0 for all admissible parameter values.

In addition, the Appendix establishes:

φ ψ φ ψ β ρ γπ π π π πxw

ww

w wa a a a= −( ) = − −( )+ −( )⎡⎣ ⎤⎦1 1 1, , (3.21)

where

ψ γρβ ρ γ ρπ

ww

wa=

−( ) −( )+ −( )1 1 1.

According to (3.21), the sign of Kxw is definitely negative. To see why,

consider a scenario in which the period t state of technology, at , is fixed and a signal arrives that at+1 will jump. That this can be ex-pected to create expected deflation can be seen by considering the extreme case in which the nominal wage rate is literally fixed. In this case, constancy of wt and wt �1requires that an x-percent increase in


technology be accompanied by an x-percent decrease in the contem-poraneous price level. This implies that the current price level re-mains fixed after a 1-percent shock to ]t

1 , while the period t + 1 price level falls by 1 percent, i.e., Ut+1< 0: This anticipated deflation, under a price inflation-targeting rule with aU> 1, is met in a fall in Rt suf-ficiently large so that the real interest rate also falls. This expansion-ary monetary reaction raises the period t output gap by stimulating period t spending. The wealth effect associated with the anticipated future rise in technology also helps to drive up spending.

By (3.20), the impact on period t price inflation, Ut, of a signal, ]t

1 , about future technology corresponds to φπw . As in the case of

sticky prices, the sign of φπw is ambiguous (3.21). Present consider-

ations alone (i.e., G = 0) make it positive. This is because the monetary expansion described in the previous paragraph increases the current marginal cost of working, and this places upward pressure on Uw,t ac-cording to the wage Phillips curve (3.15). Considerations of the fu-ture alone make φπ

w negative. Intuitively, wage inflation in the next period, Uw,t+1, can be expected to fall with the anticipated jump in at+1 because of the negative sign ofηπ

w (Proposition 3.2). The nature of the Calvo-style wage frictions suggest that Uw,t should fall in anticipation of the fall in Uw,t+1 (3.15). To determine the sign of φπ

w for interesting values of the parameters requires numerical simulation.

Departing momentarily from our main theme, we note that in the pure sticky wage model, a monetary policy that relates the nominal rate of interest to price inflation does not optimize social welfare. As em-phasized by EHL, the efficient allocations can be supported by a rule that replaces price inflation in the interest rate-targeting rule with wage inflation. To see this, note that in this case the equilibrium conditions formed by the wage-targeting interest rate rule, (3.15) and (3.18), do not include the natural rate of interest. As a result the variables, x t , Rt

and ˆ

,Uw t determined by those equations evolve independently of the technology shock. In particular, the first-best outcomes,

ˆ ˆ,xt w t= =π 0 ,

and Rt "0 satisfy the equilibrium conditions with wage targeting. According to (3.20), the rate of price inflation, Ut, is the negative of


technology growth under a wage-targeting monetary policy. Because the nominal wage rate is constant under this monetary policy, while the real wage must fluctuate with technology, it follows that optimal policy does not stabilize the high frequency movements in inflation in the pure sticky wage case.

3.3. Numerical Results

In this section, we report numerical simulations of the period t impact on inflation and output of a signal,]t

1 , that technology will expand by 1 percent in the next period. To investigate robustness of the analysis, we embed the time series representation of at in (3.1) in the following more general representation:

a a u ut t t t t t= +( ) − + ≡ + ≤− −ρ λ ρλ ξ ξ ρ λ10

11 1, , | |,| | . (3.22)

The representation in (3.1) corresponds to (3.22) with Q = 0: When W + Q > 1, then (3.22) implies at follows a “hum-shape” pattern after an innovation to at. As indicated in our discussion of sticky prices, with Q sufficiently large, the model is expected to predict a rise in inflation in the wake of a positive signal,]t

1. Numerical results are re-ported in Table 5, and the value of Q is indicated in the first column. Results for the forward-looking rule, (3.4), are reported in Panel A of the table. As a further check on robustness, we also report results for the case where the interest rate responds to the contemporaneous rate of inflation, rather than to its expected value in the next period. Results for this case are presented in Panel B. We adopt the following baseline parameterization of the model:

G�"��¼, aU�"��]w= ]p= 0.75, Q�"��Q�"��XL=1, Q\"��

In the pure sticky price version of the model, ]w =0 and ]p =0.75, while in the pure sticky wage version, ]w =0.75 and ]p = 0.

Consider first the results for sticky prices in Panel A. Note that in the case stressed in the text, Q = 0, inflation falls 2.8 basis points in the period that ]t

1 jumps by 0.01, or 1 percent. At the same time, employment jumps by nearly 1 percent and the nominal rate of interest falls by 29 basis points. Under the efficient monetary poli-cy, the interest rate jumps a full 100 basis points, employment does not change and inflation remains at zero. Evidently, the interest


rate-targeting rule that feeds back on expected inflation produces very inefficient results. It creates a boom where there should be none, and it does not stabilize inflation.

Note that as Q increases, the interest rate-targeting rule becomes more inefficient. For the largest value of Q considered, employment increases 2.5 percent in the period of the signal shock. However, the cases cease to be relevant from an empirical standpoint because infla-tion now increases in response to the signal shock.

Motivated by the fact that equilibrium models that do well empiri-cally also incorporate sticky wages, we now consider the sticky wage case in Table 5. Note that with sticky wages, inflation is predicted to fall and output rise for all the values of Q reported. Thus, while the model with sticky prices is not robust to a hump-shape representation

Table 5Period t Response to News, ]t

1 , that Period t + 1 Technology Innovation Will be 1 Percent Higher

In all cases, the natural rate, Rt�, jumps 100 basis points

at = (0.9 + Q) at-1_0.9Q�Ft-2 +ξ ξt t

01

1+ −

Panel A: Policy rule - ˆ ˆR a Et t t= +π π 1

Q Utht Rt EtRt/ Ut+1

Sticky P Sticky W Sticky P Sticky W Sticky P Sticky W Sticky P Sticky W

0 -2.8 -15 0.98 0.84 -29 -175 -9.8 -58

0.125 -0.42 -16 1.1 0.98 -29 -178 -9.8 -59

0.2 1.7 -18 1.2 1.1 -29 -181 -9.6 -60

0.6 42 -28 2.1 2.3 8.2 -200 2.8 -67

0.8 117 -25 2.5 4.7 111 -206 37 -69

Panel B: Policy rule - ˆ ˆR at t= ππ0 -1.7 -10 0.78 0.93 -2.5 -15 13 97

0.125 -0.25 -11 0.87 1.1 -0.4 -16 15 97

0.2 1.0 -12 0.93 1.2 1.5 -17 16 97

0.6 22 -16 1.4 2.3 33 -24 35 98

0.8 61 -8.7 1.6 4.2 92 -13 58 106

Notes: (1) Inflation and rates of return, Ut, Rt, EtRt/Ut+1 expressed in deviations, in units of quarterly basis points, from steady state. (2) Hours worked, ht, is expressed in percent deviation from steady state. (3) For parameter values, see text.


Chart 5Contemporaneous Effects on Output Gap (Kx��and Inflation (KU��

of a 1-percent Signal Shock

of at, one that also incorporates sticky wages can be expected to predict more robustly that inflation falls and the output gap rises in response to a signal shock.

Now consider Panel B, which reports results for the contempo-raneous specification of the interest rate rule. The results for Ut, ht, Rt are qualitatively similar to the results in Panel A. Chart 5 reports the period t impact on the output gap (Kx) and inflation (KU) of a 1-percent news shock under perturbations to our baseline model pa-rameterization. In each case, we fix Q = 0 and use the policy rule in (3.22). In addition, the parameter perturbations reported change only the value of the parameter indicated and hold the other param-eters at their baseline value. As in Table 5, the sticky wage model is more robust in predicting KU < 0. For example, if the price stickiness parameter, ]p, falls substantially below the benchmark value, then KU�> 0. However, KU < 0 for all values of ]w reported. Similarly, if aU


is substantially above its value in the benchmark parameterization, then KU� > 0 with pure sticky prices, but KU� < 0 with pure sticky wages. Finally, Kx > 0 for all parameterizations considered.

In summary, our benchmark sticky price model predicts that infla-tion drops and employment rises in the period that a signal about a future technology expansion arrives. This resembles the pattern ob-served for stock market booms. When we depart substantially from the benchmark parameterization, the model predicts a rise in inflation after a news shock. However, a model with pure sticky wages predicts much more robustly that inflation drops and the output gap rises in response to a signal shock. We conclude that models with sticky wages and sticky prices are likely to robustly predict that inflation falls and output rises in response to a signal shock. Models that simultaneously incorporate both sticky wages and sticky prices have a state variable and are not so easy to solve analytically as the case of pure sticky wages and pure sticky prices considered here. We turn to the model that in-corporates both sticky wages and prices in the next section.

4. Analysis in a Medium-Sized Model

In this section, we consider a medium-sized New Keynesian model fit to U.S. postwar data by Bayesian methods.29 Relative to the ma-terial in the previous section, the analysis here has the disadvantage that it cannot be done analytically. On the other hand, the results may perhaps be taken more seriously because they are produced in a model that generates time series data that more closely resemble actual U.S. data. In addition, in this model, we are able to consider the impact of optimistic expectations about the future on the stock market (however, the model shares the shortcoming of most models in that it understates the magnitude of volatility in the stock market). The stock market is a variable that is missing in the analysis of the previous section. Finally, by adding the financial frictions proposed in BGG to our estimated model, we are able to consider interesting modifications to the inflation forecast-targeting interest rate rule. We find that when we allow credit growth to play an independent role in that rule, one that goes beyond its role in forecasting inflation, then the interest rate-targeting rule’s tendency to produce excessive volatility in response to optimistic expectations about the future is


reduced. We interpret this as evidence that credit growth is correlated with the natural rate of interest. The natural rate of interest is what one really wants in the interest rate-targeting rule, and credit growth appears to be a good proxy, at least relative to shocks to expectations about the future.

4.1. A Medium-Sized Model

The estimated model incorporates Calvo-style sticky prices and wages, habit persistence in preferences, variable capital utilization and adjustment costs in the change in investment. We do not display the shocks that were used in the estimation of the model. This sec-tion presents simulations of the model analogous to the simulations performed in the previous section. Our presentation of the model is limited to what is relevant for those simulations. As in EHL, we suppose that households supply a differentiated labor service, lt,j, j �(0,1). Preferences of the household supplying the jth type of labor services are given by:

E C Cl

tj l

t l t lt l jβ log . ,

+ + −+−( )−

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪0 75 110

21

2

,, . ,/β = −

=

∞

∑ 1 03 1 4

0l

where Ct denotes consumption. The household is a monopoly sup-plier of its type of labor service and sets the wage rate, Wjt, subject to the demand for lt,j and to the following friction. With probability, ]w = 0.80, the household cannot reoptimize its wage and with the complementary probability, it can set the wage optimally. In case it cannot reoptimize its wage, Wjt is set as follows:

W Wjt z jt= −πµ 1,

where Rz = 1.0038 is the steady-state growth rate of the underlying shock to technology and U = 1.006 is the steady-state rate of inflation. The household accumulates capital subject to the following technology:

K K SI

IIt t

t

tt+

−

= − + −⎛

⎝⎜

⎞

⎠⎟1

1

1 1( ) ( ) ,δ


where K tis the beginning of period t physical stock of capital, and It is

period t investment. The function S is convex, with S (Rz) = S (Rz) = 0 and S "(Rz) = 2.2. The physical stock of capital is owned by the house-hold and it rents capital services, Kt, to a competitive capital market:

K u Kt t t" ,

where ut denotes the capital utilization rate. Increased utilization requires increased maintenance costs in terms of investment goods according to the following function:

a u Kt t( ) ,

where a is increasing and convex, a (1)= a' (1) = 0, a" (1) = 0.02 and ut is unity in nonstochastic steady state.

The households’ specialized labor inputs are aggregated into a ho-mogeneous labor service according to the following function:

L l dit t i ww

w

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=∫ ( ) , . .,

1

0

11 05λ

λ

λ

A final good, Yt, is produced by a representative, competitive firm according to the following technology:

Y Y dlt lt ff

f

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

=∫ ( ) , . .1

0

11 20

λ

λ

λ

Here, Ylt is the l th intermediate good produced by a monopolist us-ing the following technology:

Y z A L K z tl t t t l t l t t z, , ,( ) ( ), exp( ), . ,= = =−1 0 4α α µ α

where Kl,t, Ll,t denote the capital and labor services used by the lth monopolist. Also, at = log (At) and has law of motion analogous to the one in (3.1):

a at t t t= + +− −0 9 10

88. ,ξ ξ


where]t0, ]t �8

8 are iid shocks that are uncorrelated with each other at all leads and lags, and with at-j , j > 0. The shock, ξt i

i− , is observed by

agents at date t-i. We refer to ]t �88 as a “signals” about at that arrives

eight quarters in advance.

The monopoly supplier of the intermediate good can reset its price optimally with probability 1–]p, ]p= 0.77 and with probability ]p it follows the following simple rule:

P Pl t l t, , .= −π 1

Monetary policy is governed by the following interest rate rule:

log log ,RR

RR

Rt tt

⎛

⎝⎜

⎞

⎠⎟=

⎛

⎝⎜

⎞

⎠⎟+ −( )−� �ρ ρ1 1 (4.1)

where Rt denotes the gross nominal rate of interest and

R a Ea y

yt tt y t=

⎛

⎝⎜

⎞

⎠⎟+

⎛

⎝⎜

⎞

⎠⎟+

πππ

log log ,1

4 (4.2)

where aU = 2.25, ay = 0.32: Here, yt denotes GDP (scaled by zt) and y denotes the corresponding steady-state value. Also, �P = 0.57.

4.2. Simulation

Chart 6 presents the results of simulating a particular stock market boom-bust episode. In the first period, a signal,ξt

8 0> , arrives, which creates the expectation that at two years later will jump. However, that expectation is ultimately disappointed, because ξt +8

0 = –ξt8

Thus, in fact nothing real ever happens. The dynamics of the econo-my are completely driven by an optimistic expectation about future productivity, an expectation that is never realized. This experiment has a variety of interpretations. One is that people receive actual evi-dence that things will improve in the future, evidence that ultimately turns out to be false. Another is that they are irrationally optimistic about the future, and they realize their error when the thing they expected does not happen.


In interpreting the results, it is important to recognize that whether or not the signal is realized is irrelevant for the analysis in the periods before the anticipated event is supposed to occur. This is true wheth-er we consider optimal policy, or policy that sets the interest rate ac-cording to a particular rule. This is because neither actual policy nor the policymaker implementing the optimal policy makes use of any information beyond what private agents know.

We simulate the dynamic response of the economy under two cir-cumstances. The thin line in Chart 6 corresponds to the response of the baseline model, the one defined in the previous subsection. The starred line corresponds to the response in the Ramsey-efficient equilibrium corresponding to the baseline model. To obtain the Ramsey equilibrium, we drop the monetary policy rule. The system is now undetermined, there being many constellations of stochas-tic processes that satisfy the remaining equilibrium conditions. The Ramsey equilibrium is the stochastic process for all the variables that

Chart 6Response of Baseline and Perturbed Model to Signal Shock

(signal not realized);Perturbation = Ramsey


optimizes a social welfare criterion constructed by integrating the utility of each type j household, j �(0, 1). The Ramsey equilibrium roughly corresponds to the equilibrium associated with the real busi-ness cycle model obtained by shutting down the wage- and price-set-ting frictions and by imposing that all intermediate good firms pro-duce at the same level and each type j worker works the same amount. We say “roughly” here because deleting only one equation (i.e., the interest rate rule) does not provide enough degrees of freedom for the Ramsey equilibrium to literally extinguish all the frictions in the model. The Ramsey equilibrium forms a natural benchmark because it corresponds to the equilibrium with optimal monetary policy.

Note how the rise in investment, consumption, output and hours worked in the baseline equilibrium exceeds the corresponding rise in Ramsey equilibrium by a very substantial amount. This excess en-tirely reflects the suboptimality of the monetary policy rule, (4.1). Interestingly, the inflation rate in the boom is below its steady-state value of roughly 2.5 percent annually, as in the examples of the previ-ous section and as in the data. At the end of the boom, inflation rises a bit. According to the evidence in Adalid and Detken (2007), this is what typically happens in boom-bust episodes: Inflation is low in the boom phase and then rises a little at the end.

The 3,1 panel in the chart displays the response of the price of cap-ital in terms of consumption goods in the model. We interpret this as the price of equity in the model. Note that in the baseline model, the price of capital rises during the boom. In the Ramsey equilibrium, the price of capital actually falls. One way to understand this fall in the price of capital is that the real interest rate in the Ramsey equi-librium (the “natural rate of interest,” in the language of the previous section) rises sharply with the signal shock. The increased discount-ing of future payments to capital explains the fall in the price of capital.30 Monetary policy in the baseline equilibrium prevents the sharp rise in the interest rate. This is the heart of the problem with the monetary policy rule. The interest rate should be raised, as in the Ramsey equilibrium, but there is nothing in the monetary policy rule that produces this outcome. The most important variable in the in-terest rate-targeting rule, inflation, actually drives the interest rate in


the wrong direction. In effect, monetary policy is overly expansion-ary in the boom. This is what makes the stock market boom (actu-ally, it is not a very strong boom) and what makes indicators of aggre-gate activity boom too. As Chart 6 indicates, only a small part of the boom reflects the operation of optimistic expectations. The boom is primarily a phenomenon of loose monetary policy. Again, it bears repeating that the nature of the boom is independent of whether the signal is ultimately realized or not.

Interestingly, an outside observer might be tempted to interpret the rise in labor productivity during the boom as indicating that an actual improvement in technology is under way. In fact, the rise in productivity reflects a sharp rise in capital utilization. This phenom-enon would be even greater if the model also incorporated variations in labor utilization.

In sum, an interest rate-targeting rule that assigns substantial weight to inflation transforms what should be a modest expansion into a significant boom. The reason is that the monetary policy does not raise the interest rate sharply with the rise in the natural rate of interest. This problem can be fixed by setting the interest rate to the natural rate or, if that is deemed too difficult to measure, to some variable that is correlated with the natural rate. We explored the lat-ter option. We added the financial frictions sketched by BGG to the baseline model in order to obtain a model in which credit growth plays an important economic role. We found that when we simu-late the resulting model’s response to a signal shock, the equilibrium more closely approximates the corresponding Ramsey equilibrium if credit growth is introduced into the interest rate-targeting rule, (4.1). In particular, we replace Rt in (4.2) with:

R Eyyt t t yt

c= ( )−⎡⎣ ⎤⎦+⎛

⎝⎜

⎞

⎠⎟++α π π α απ 1

14

log x nominall credit growtht c, . ,α = 2 5

where credit is the quantity of loans obtained in the BGG model by entrepreneurs for the purpose of financing the purchase of capital. We find that in the baseline model, with Fc = 0, households would pay 0.23 percent of consumption forever to switch to the Ramsey equilibrium. We also find that households would pay 0.19 percent


of consumption forever to switch to the equilibrium in which Fc = 2.5. We interpret this as evidence that including credit growth in the interest rate rule moves the economy a long way in the direction of the Ramsey equilibrium, in which monetary policy sets the interest rate to the natural rate of interest. These calculations have been done relative to signal shocks. A full evaluation of the policy of including credit in the interest rate-targeting rule would evaluate the perfor-mance of this change when other shocks are present as well.

5. Conclusion

We have reviewed evidence that suggests that inflation is typically low in stock market booms and credit growth is high. The obser-vation that inflation is low suggests that an interest rate-targeting rule that focuses heavily on anticipated inflation may destabilize asset markets and perhaps the broader economy as well. The observation that credit growth is high in booms suggests that if credit growth is added to interest rate-targeting rules, the resulting modified rule would moderate volatility in the real economy and in asset prices.

These inferences based on examination of the historical data consti-tute conjectures about the operating characteristics of counterfactual policies. To fully evaluate conjectures like these requires constructing and simulating a model economy. This is why we devoted substantial space model analysis. The model simulations reported in the paper lend support to our conjectures.

Authors' Note: The views expressed in this paper are those of the authors and do not necessarily reflect those of the ECB or the Eurosystem. We are grateful for discussions with David Altig, Gadi Barlevy, Martin Eichenbaum, Ippei Fujiwara and Jean-Marc Natal, and for comments from John Geanakoplos. We have also benefited from the advice and assistance of Daisuke Ikeda and Patrick Higgins.


Appendix: Deriving the Equilibrium Conditions for the Simple Model

We present a formal derivation of the simple model in Section 3. Although the results are available elsewhere, we report them here for completeness. We suppose that the law of motion for log technology, at, has the following representation:

at = Wat-1 +]t0 +ξt −1

1 .

Household preferences are:

E CL

1tl

tt l

Ll

L

βσ

σ

log – .+++

=

∞

( ) +

⎡

⎣⎢

⎤

⎦⎥∑ L

1

0

Households participate in spot competitive labor markets, where the wage rate, Wt, is set flexibly. In addition, in period t, households have access to a bond market in which the gross nominal rate of interest from t to t+1 is denoted Rt. In addition, households purchase con-sumption goods, Ct, at price, Pt. The household budget constraint is:

PtCt + Bt+1 < WtLt + Rt-1Bt + Tt,

where Tt denotes lump sum income from profits and government transfers. The first order necessary condition associated with the household’s labor supply and savings decisions are:

ψ βπ

σL t t

t

t tt

t

t

t

L CWP C

EC

RL = =

+ +

, .1 1

1 1 (A.1)

A.1. Sticky Prices

Final goods, Yt, are produced as a linear homogeneous function of Yit, i �(0, 1) using the following Dixit-Stiglitz aggregator:

Y Y dlt ltf

f

=⎡

⎣⎢⎢

⎤

⎦⎥⎥∫0

11λ

λ

.


The representative, competitive final good producer buys the ith in-termediate input at price, Pit. The ith input is produced by a monopo-list, with production function:

Yit = exp (at)Lit.

Here, Lit denotes labor employed by the ith intermediate good pro-ducer. The ith producer is committed to sell whatever demand there is from the final good producers at the producer’s price, Pit. The pro-ducer receives a tax subsidy on wages in the amount (1–S)Wt. The subsidy is financed by lump sum taxes on households. The ith inter-mediate good producer sets Pi,t subject to Calvo frictions. Thus, in any period 1– ]p�randomly selected producers may reset their price, and the remainder must set price according to:

Pi t P

P

t p

i t

, { ,= −� with probability 1-

with prob

ξ1 aability ξp

The resource constraint is:

C Yt tf .

We now describe the equations that characterize the private sector.

A.1.1. Private Sector Equilibrium

We obtain a set of equations that characterize a private sector equilibrium. We then describe a log-linearization of those equations about steady state. It is easy to verify that each of the intermediate good producers that has an opportunity to set price in period t does so as follows:

� ��

pKF

pPPt

t

tt

t

t

= ≡, .

Here, �Pt denotes the price set by an optimizing intermediate good producer. Also,

K s E K

F E F

s

t f t P t t t

t p t t t

t

= +

=

=

+ +

+−

+

λ βξ π

βξ π

ε

ε

1 1

11

11+

(1– )ν ψ σL t t

t

L CA

L

.


Here, st denotes the (after subsidy, S) marginal cost of production, where the real wage has been replaced by the household’s marginal rate of substitution between consumption and leisure (we impose a unit Frisch elasticity of labor supply).

Combining the final good production function with the first order condition of intermediate good producers implies

P P dit i tf

f

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

−

−

∫ , .1

1

0

1

1

λ

λ

Evaluating this integral, taking into account the price set by current-period optimizers and taking into account that firms that do not reoptimize price are selected randomly, we obtain (after rearranging),

�ptp t

p

f

f

=−−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

−

1

1

11

1

ξ πξ

λ

λ

.

We conclude:

KF

t

t

p t

p

f

f

=−

−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

−

−

1

1

11

1

ξ πξ

λ

λ

.

In addition, the relationship between Yt and aggregate employment is given by:

Y p Lt t t" * ,

where law of motion for the price distortion is:

pt pp t

p

p tf

f

* = −( ) −−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

+−

11

1

11

ξξ πξ

ξ πλ

λ λλλ

f

f

pt

−

−

−⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

1

1

* .


The resource constraint implies Ct = Yt.

We conclude that the equations that characterize a private sector equilibrium are:

1 1

11

1 1 11

p LE

p L a a

R

p

t tt

t t t

t

t

t

t

* *

*

exp=

−( )

= −

+ + ++

βπ

ξppp t

p

p t

t

f

f f

f

p( ) −

−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

+− −

−

1

1

11 1ξ π

ξξ πλ

λ λλ

11

1

11

1

1

1

1

*

*( )

F E F

K L p

t p t t t

t f t t

f

L

= +

= − +

+−

+

+

βξ π

λ ν β

λ

σ ξξ π

ξ πξ

λλ

λ

p t t t

t

t

p t

p

E K

KF

f

f

f

+−

+

−

=−

−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

1

1

1

11

1

1

⎥⎥⎥⎥⎥

−1 λ f

.

Here, we have used the aggregate resource relation to substitute out for consumption. We assume that the labor market subsidy is set to eliminate the monopoly inefficiency in the labor market. That is,

Qf (1– S) = 1.

Thus, there are five equilibrium conditions in six variables: Lt, Rt, pt*,

Ft, Kt, Ut.

For a given value of steady-state inflation, U, we use the steady-state version of the above equations to solve for the steady-state values of the other variables. In steady state, the five equations that characterize private sector equilibrium reduce to the following:


R p

p

p

p

p

f

f

f

= =

−−

− ( )−

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

−

−

πβ

ξ πξ

ξ πξ

λλ

λ

, *

1

1

1

1

1

11 ⎟⎟⎟⎟⎟

=

−

=−

−−

+

λ

λ

σ

λ

βξ π

ν λ

βξ π

f

f

L

fF K

L p

p t

f

p

1

1

1

1

11

1

,( ) *

λλ

λ

λ

ξ πξ

f

f

f

K F p

p

−

−

−

=−

−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

11

1

1

1.

The first three equations allow one to compute R, p*, F. Combining the last two:

1

1

1

1

1

1

11−( )

−

=−

−

⎡

⎣

⎢⎢⎢

+

−

−ν λ

βξ π

ξ πξ

σ

λλ

λf

p

p

p

L pF

L

f

f

f*

⎢⎢

⎤

⎦

⎥⎥⎥⎥

−1 λ f

,

which can be solved for employment, L:

Lp

Fp

f

p

p

f

f f

2

11

11

1

1

1=

−−

−−

⎡

⎣

⎢⎢⎢⎢

⎤− −βξ π

ν λξ π

ξ

λλ λ

( ) *

⎦⎦

⎥⎥⎥⎥

=−

−

−

−

−

1

1

1

11

1

1

λ

λλ

λν λβξ π

βξ π

f

f

f

ff

p

p

p( ) *−−

−

−

−−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

1

11

1

1

1

ξ πξ

λ

λ

p

p

f

f

.


Finally, K is computed using one of the two equations above in K.

Let z t denote dzt ⁄z, for a small deviation, dzt = zt–z. Totally differ-entiating the five equilibrium conditions about steady state, we obtain:

− +( ) = − − − − −+ + + +ˆ ˆ ˆ ˆ ˆ ˆ ( )* *p L R p L da dat t t t t t t tπ 1 1 1 1

ˆ *ptf p

f

p

p

f

f

=−

−−

⎛−

−λ ξλ

πξ π

ξλ

λ

1

1

1

11

11

⎝⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

−

⎡

⎣

⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥

+

−

−

λλλπ π ξ π

f

f

fp pt p

1

1* ˆtt

t p tf

t tE F

−

+ +=−

+

1

1 1

11

*

ˆ ˆ ˆ(1)F βξλ

π⎡⎡

⎣⎢⎢

⎤

⎦⎥⎥

= −( ) +( ) +(2)K ˆ ˆt p L tL1 1βξ σ pp E Kt p t

f

ft t

* ˆ ˆ⎡⎣

⎤⎦+ −

+⎡

⎣⎢⎢

⎤

⎦⎥⎥+ +βξ

λλ

π1 1 1

(3)K ˆ ˆt

p

pt tF=

−+

ξξ

π1

Substitute out for K t from (3) into (2), and then (1) into the result, to obtain:

ˆ ˆ ˆ

ˆ ( ) ˆ ˆ ,

( )(

*

π κ βπσ

κβξ ξ

t t t

t L t t

p

s

s L p

= +

= + +

=− −

+1

1

1 1 pp

p

),

ξ

(A.2)

where st denotes the percent deviation of real marginal cost from its steady state. Note that when U = 1, then the law of motion for the

price distortion simplifies to ˆ ˆ* *p pt p t= −ξ 1 . This implies that, as long

as the system has been operating for some time, we can simply set ˆ*pt = 0 for all t. In this case, the linearized intertemporal equation can be written:

− = − − − −+ + +ˆ ˆ ˆ ˆ ( )L R L a at t t t t tπ 1 1 1 ,


where we have dropped the “d” in front of the at’s because the steady state cancels.

We take the efficient equilibrium to be the first-best, in which inflation is zero, Lt = 1 and Ct = At. It can be shown this is the Ramsey efficient equilibrium when the labor subsidy is chosen to eliminate the distortions of monopoly power in the steady state and price dis-tortions are zero in the initial period.

A.1.2. Interest Rate Rule Equilibrium

We assume the monetary authority implements the following monetary policy rule:

log log( ),RR

a Ett t

⎛

⎝⎜

⎞

⎠⎟= +π π 1

where the implicit target for inflation is its efficient level of zero. Substituting these assumptions into the monetary policy rule and linearizing the latter about steady state, we obtain:

ˆ ˆ .R a Et t t= +π π 1

Equilibrium output is yt = exp (at)pt* Lt. In a zero inflation steady

state L =1 and we may, to a first approximation, pt* = 1. Thus, xt, the

ratio of actual output to efficient output is:

xa p L

aLt

t t t

tt= ( ) =

exp

exp( )

*

and

ˆ ˆ ,x Lt t"

where xt is the percent deviation of actual output from Ramsey out-put. With these modifications, the intertemporal Euler equation can be written:

ˆ ˆ ˆ ˆ .*x R R xt t t t t= − − −⎡⎣

⎤⎦++ +π 1 1


Here, xt is the output gap.

We now explain the formulas for the solution to the simple model in Section 3. Substituting out for Rt andRt

� in the IS curve from the policy rule and the Rt

� equation, we obtain:

ˆ ˆ ( ) ˆ

ˆ

x E a a E xt t t t t t t

t

= − −( ) − − −⎡⎣ ⎤⎦+

=+ +π π ρ ξ

π γ

1 111

1

ˆ ˆ ,x Et t t+ +β π 1

where,

L�} P�(1 + XL).

Substituting in the posited solution,

η φ ξ η ρ ξ ρ ξπ πx t x t t t t ta a a a+ = − −( ) +⎡⎣ ⎤⎦− −( ) −⎡⎣

⎤⎦

1 1 11 1 ++ +⎡⎣ ⎤⎦

+ = +⎡⎣ ⎤⎦+

η ρ ξ

η φ ξ γ η φ ξ βη ρπ π π

x t t

t t x t x t

a

a a a

1

1 1tt t+⎡⎣ ⎤⎦ξ1 .

Collecting terms in at and ]t1 :

η η ρ ρ η ρ

φ ηπ π

π π

x x

x

a

a

= − −( ) − −( )⎡⎣ ⎤⎦+

= − −( ) −⎡⎣ ⎤⎦+

1 1

1 1

,

ηη

η γη βη ρφ γφ βη

π π

π π

x

x

x

= += +

,

.

Using the first of these equations to simplify the second, we obtain:

η η ρ ρ η ρ

φ ηρ

η γη βη ρφ γ

π π

π π

π

x x

xx

x

a= − −( ) + − +

= +

= +=

1 1

1

,

,

,

φφ βηπx + .


Solving the third equation:

η γρβ

ηπ = −1 x .

Using the latter to solve the first:

ηγρβ

ρρπ

x

a= −

+ −( ) − −

1

1 11 1

.

Now, consider KU :

φ γφ βη γ ηρ

β γρβ

η

γρ ρβ

π π

π

= + = + +−

−− + −( )

xx

x

a

11

11

1 1=

γγρρ

γβ ρ γρβ

π

1

1 1

1 1

−

⎧

⎨⎪⎪

⎩⎪⎪

⎫

⎬⎪⎪

⎭⎪⎪

− −( )+ −( )−( )=

a

−−( )+ −( )ρ γρπa 1.

A.2. Sticky Wages?

In this section, we interpret the labor, Lt, hired by intermediate good firms as supplied by “labor contractors.” These contractors sup-ply Lt by combining a range of differentiated labor inputs, ht,j , using the following technology, or “aggregator function”:

L h djt t j ww

w

= ( )⎡

⎣⎢

⎤

⎦⎥ ≤∫ , , .

1

0

11λ

λ

λ

The labor contractors are perfectly competitive. They take the wage rate, Wt, of Lt as given. They also take the wage rate, Wt,j , of the jth

labor type as given. Contractors choose inputs and outputs to maxi-mize profits,


W L W h djt t t j t j− ∫ , , .0

1

The first order necessary condition for optimization is given by:

hWW

Lt jt

t jt

w

w

,,

.=⎛

⎝⎜⎜

⎞

⎠⎟⎟

−λλ1

(A.3)

Substituting the latter back into the labor aggregator function and rearranging, we obtain:

W W djt t j

w

w

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

−

∫−

,

11

0

1

1λ

λ

. (A.4)

We now turn to the households. We adopt the “indivisible labor.” Accordingly, suppose there is a large number of identical households. Each household has many members corresponding to each type, j, of labor. Each worker of type jth has an index, l, distributed uniformly over the unit interval, [0, 1], which indicates that worker’s aversion to work. A type j worker with index l experiences utility:

log (Ct)– 1XL, XL > 0,

if employed and,

log (Ct ),

if not employed. The notation reflects that each worker in a house-hold, whether employed or not and regardless of labor type, enjoys the same amount of consumption. This is the efficient arrangement, given our assumption that worker utility is separable in consumption and leisure and the household objective is to maximize the equally weighted integral of worker utility.

The quantity of labor supplied by the representative household, ht,j , is determined by (A.3). We suppose the household sends j-type workers with 0 < l < ht,j to work and keeps those with l > ht,j out of the labor force. The equally weighted integral of utility over all l �[0, 1] workers is:


Demand for labor, ht,j , is determined by a monopoly union.

log ,Ch

tt j

L

L

( )− +

+1

1

σ

σ

Overall household utility also integrates over all j–type workers:

log .,Ch

djtt j

L

L

( )− +

+

∫1

0

1

1

σ

σ (A.5)

It remains to explain how Ct and ht,j are determined.

The wage rate of the jth-type of labor, Wt,j, is determined outside the household by a monopoly union that represents all j-type work-ers across all households. The union’s problem is discussed below. The household seeks to maximize the expected present discounted value of utility, (A.5), subject to the following budget constraint:

PC B B R W h djt t t t t t j t j+ ≤ + ++ − ∫1 1 0

1

, , Transfers and prrofitst . (A.6)

The only thing for the household to do is choose Ct and Bt+1. The first order necessary condition for optimization implies (A.1).

For each j there is a monopoly union that represents all type j workers across all households. The union is required to satisfy its demand curve, (A.3). It faces Calvo frictions in the setting of Wt,j. With probability 1–]w, a union can optimize the wage and with the complementary probability, ]w, it cannot, in which case,

Wt,j = Wt–1,j. (A.7)

With this specification, the wage of each type j of labor is the same in the steady state. Because the union problem has no state variable, all unions with the opportunity to reoptimize in the current period face the same problem. In particular, such a union chooses the current value of the wage,Wt , to maximize: ~


E W hh

t wi o

i

t i t t it t i

t

L t i

L

βξ υσ υ

σ

( ) − ( )+( )=

∞

+ ++

+

+∑ !

1

1

⎡⎡

⎣

⎢⎢

⎤

⎦

⎥⎥.

(A.8)

Here, St+i denotes the marginal value assigned by each household (recall, they are all identical) to the wage.31 The household treats Zt as an exogenous constant. In the above expression, ]w appears in the discounting because the union’s period t decision only impacts on future histories in which it cannot reoptimize its wage. Also, ht i

t�

denotes the level of employment in period t+i of a union that had an opportunity to reoptimize the wage in period t and did not reopti-mize again in periods t +1, …, t + i. By (A.3),

hW

WL

wt it t

t it i

t

w t w t i

w

w

++

−

++ +

=⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝

!

…

λλ

π π

1

1, ,

⎜⎜⎜⎞

⎠⎟⎟

−

+

λλw

w

Lt i

1

,

(A.9)

where

wWW

wW

a Ptt

tt

t

t tt i a t a t a

≡ ≡ ( ) ≡+ +

!…,

exp, , , ,χ µ µ µ

1

11 2 ,,

.t i t t i

ii+ + +

⎧⎨⎪

⎩⎪≥=π π1

10

… (A.10)

In (A.10), Ra,t represents the growth rate of technology:

µa tt

t

aa,

exp( )exp( )

.=−1

Using (A.9) to substitute out for ht+it in (A.8):

E WW

WL

W

t wi

i

t i tt

t it i

tw

w

βξ υ

λλ

( )⎛

⎝⎜

⎞

⎠⎟ −

=

∞

++

−

+∑o

!!

!1 WW

Lt i

t i

L t i

w

wL

L

+++

+

⎛

⎝⎜

⎞

⎠⎟

+( )

⎡

⎣

⎢⎢⎢⎢

−+( )λ

λσ

σ

σ υ

11

1

1⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

.

Differentiating with respect to Wt

, we obtain, after rearranging,


Ehc

w w MRSt wi o

it it

t it t t i w t i

tβξ χ λ( ) −⎡⎣ ⎤⎦=

∞+

++∑ , ,

(A.11)

where,

cC

a P Ctt

tt j

t j t j

≡ ( ) =++ +exp

, .ν 1

(A.12)

Also, MRStt+i in (A.11) denotes the (scaled) cost of working for the

marginal worker in period t + i whose wage was reoptimized in pe-riod t and not again reoptimized in periods t + 1, t + 2, … t + i :

MRSh

a Pc ht i

t t it

t i t i t it i t i

t

L

++

+ + ++ +≡ ( )

( ) =σ

σ

νexp( ) LL .

According to (A.11), the union seeks to set the wage to a markup, Qw, over the cost of working of the marginal worker, on average.

We now expand (A.11) about a steady state in which:

w i w MRSt i t w t w t= = = = =1 1 1 1, , , , , .,allχ λ π π

It is convenient to obtain some preliminary results. Note,

ˆ ˆ ˆ ˆ ˆ ˆ, , , ,

χ µ µ πt i a t a t a t iu

t= − + + +( ){ − − −+ + + +1 2 1

ππ

σ σ

t i

t i

t

t i L L t it

t iMRS c H h Ht i

+

+ + + += + + −( )+ˆ ˆ ˆ ˆ ,

0

and, from (A.9),

ˆ ˆ( ˆ ˆ ˆ ), ,

h H

w

t it

t i

w

wt w t w t i

w+ +

+ +

− =−

− −

−

λλ

π π

λλ

1

1

1$

wwtw

ii

ˆ.

=>

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

00

i

i

>=

0

0,


We can write the discounted sum of the marginal cost of working as follows:

S MRS SMRS t wi

i

t i

t

o t Lw

w w, ,

ˆ≡ ( ) = +− −=

∞

+∑ βξ σ λλ βξ0 1

11

! ww St w t−⎡⎣ ⎤⎦, .

(A.13)

Here,

S c H c H So t w

i

t i L t i t L t w o t, ,ˆ ˆ ˆ ˆ≡ ( ) +⎡⎣

⎤⎦= + ++ + +βξ σ σ βξ 11

0i=

∞

∑ (A.14)

S Sw t w

i

w t i w w ti

w w t, , , ,ˆ ˆ .≡ ( ) = ++ +

=

∞

+∑ βξ π βξ π βξ11

1

(A.15)

The following expression is also useful:

S t w

i

t ii w

w

i

a t i tχ βξ χβξ

βξ µ π, , ,ˆ ˆ ˆ= ( ) =

− ( ) − −=

∞

+ +∑1

11 ii

i

w

wa t t

⎡⎣ ⎤⎦

−− −⎡⎣ ⎤⎦+

=

∞

+ +

∑1

1 11=

βξβξ

µ π βξˆ ˆ, ww tSχ , .+1

(A.16)

Because the object in square brackets in (A.11) is zero in steady state, the expansion of (A.11) does not require expanding the expres-sion outside the square bracket. Taking this andw MRSw= λ into ac-count, the expansion of (A.11) is:

01

1=

−+( )+ −

βξ χw

t t t MRS tw w S Sˆ ˆ ., ,

(A.17)

We now deduce the restriction across wages implied by (A.4). Us-ing (A.7) and the fact that non-optimizing unions are selected at random, (A.4) reduces to:

W W Wt w t w tw w

w

= −( )( ) +⎡

⎣⎢⎢

⎤

⎦⎥⎥

−−

−

−

11

11

11

1

ξ ξλ λ

λ

� ( ) .

Divide by Wt and use (A.10):


1 111

1

11

= −( )( ) +⎛

⎝⎜⎜

⎞

⎠⎟⎟−

−

ξ ξπ

λλ

w t ww t

w w

w

,

.

Log-linearize this expression about steady state to obtain:

ˆ ˆ .,wtw

ww t=

−ξξ

π1

Replace SMRS,t in (A.17) using (A.13) and then substitute out forwt using the previous expression:

S

11

11 1

1

−+

− −+

= +

βξ βξξξ

π

σ λ

χw

tw

w

ww t t

o t Lw

w Sˆ ˆ, ,

, −− − −−

− −λ βξξξ

π σ λλ βξw w

w

ww t L

w ww tS

11 1 1

11

ˆ, ,

w

(A.18)

Multiply (A.18) evaluated at t + 1 by G]w�and subtract the result from (A.18) evaluated at t to obtain:

11

11 11−

−( )+ − −−+βξ

βξβξ

ξξ

π βξ πw

t w tw

w

ww t ww wˆ ˆ ˆ ˆ

, ww t t w t

o t w o t Lw

S S

S S

, , ,

, ,

+ +

+

( )+ −( )

= −( )+

1 1

1

χ χβξ

βξ σ λ11

11 1

11

1

1− − −−( )

−−

+λ βξξξ

π βξ π

σ λλ

w w

w

ww t w w t

Lw

w

ˆ ˆ, ,

−−−( )+βξβξ

ww t w w tS S, , .1

Simplify this expression using (A.14), (A.15) and (A.16):

11

11 11−

−( )+ − −−+βξ

βξβξ

ξξ

π βξ πw

t w tw

w


, ww t

w

wa t t

t L tc H

,

,ˆ ˆ

ˆ ˆ

+

+ +

( )

−−

+( )

= + +

1

1 11βξβξ

µ π

σ σσ λλ βξ

ξξ

π βξ π

σ λ

Lw

w w

w

ww t w w t

Lw

11

1 1

1

1− − −−( )

−−

+ˆ ˆ

, ,

λλ βξβξ π

w ww w t

11 1− +

ˆ,

(A.19)


The relationship between wage and price inflation, the change in the real wage and technology growth is given by:

ˆ ˆ ˆ ˆ ˆ ., ,w wt t w t t a t= + − −−1 π π µ (A.20)

Use this to substitute out for πt+1 in (A.19):

11

11 11−

−( )+ − −−+βξ

βξβξ

ξξ

π βξ πw

t w tw

w


, ww tw

wt w t

t L

w w

c

t, ,ˆ ˆ ˆ

ˆ ˆ

+ +( )− −+ −( )

= +

+1 11 1

βξβξ

π

σ HHt Lw

w w

w

ww t w w t L+

− − −−( )−+σ λ

λ βξξξ

π βξ π σ λ1

11 1 1

ˆ ˆ, ,

ww

w ww w t1

11 1− − +λ βξ

βξ π , .

Collecting terms inWt ,wt �1 , ˆ,Uw t ,πt+1

:

ˆ ˆ ˆ ˆ,w c Ht

w

w

wL

w

ww t t L t+

− −−

−⎛

⎝⎜

⎞

⎠⎟ = +1

1 11

1βξξξ

σ λλ

π σ

++− −

+ −− −

+⎡

⎣⎢

⎤

⎦⎥

⎛

⎝⎜⎜

⎞

⎠

βξβξ

ξξ

σ λλ

ξξ

w

w

w

wL

w

w

w

w1 11

1 11 ⎟⎟⎟ +

ˆ,πw t 1

or, after simplifying,

ˆ ˆ ˆ ˆ,w c Ht

w

w

wL

w

ww t t L t+

− −−

−⎛

⎝⎜

⎞

⎠⎟ = +1

1 11

1βξξξ

σ λλ

π σ

++− −

−−

⎛

⎝⎜

⎞

⎠⎟ +

βξβξ ξ

σ λλ

πw

w wL

w

ww t1

11

11 1

ˆ .,

Let

κξ βξβξw

w w

w

=−( ) −( )1 1

.

Multiply the previous expression by Pw and rearrange:

11

11

−−

⎛

⎝⎜

⎞

⎠⎟ = + −( )+ −σ λ

λπ κ σ σ λ

Lw

ww t w t L t t L

wc H wˆ ˆ ˆ ˆ, −−

⎛

⎝⎜

⎞

⎠⎟ +λπ

ww t

ˆ ., 1


Divide both sides by the coefficient on ˆ,Uw t , we obtain the wage

Phillips curve:

ˆ ˆ ˆ ˆ ˆ ., ,π κ

σ λλ

σ βπw tw

Lw

w

t L t t w tc H w=+

−

+ −( )+ +

11

1

(A.21)


Endnotes1We exclude the Civil War and World War I and II from our data set.

2See also Christiano, Motto and Rostagno (2007). The position is often labelled “the BIS view” (see, Eichengreen, 2009, and White, 2009, and the references therein). For an application of the Christiano, Ilut, Motto and Rostagno (2008) analysis to the Japanese economy, see Christiano and Fujiwara (2006). In related work, Cecchetti, Genberg, Lipsky and Wadhwani (2000) argue that monetary policy should also react to movements in the stock market.

3We are sympathetic to the idea (see, for example, Hirtle, Schuermann, and Stiroh, 2009) that a macro-prudential perspective ought to be brought to bear on the supervision and regulation of financial forms. See, for example, Barlevy (2008), Evans (2010) and Geanakoplos (2009).

4In developing the intuition for our model results, we stress the case where the interest rate rule is a function of expected future inflation. However, we show that the results are robust to the assumption that the interest rate rule is a function of current, actual inflation.

5Barsky and Sims (2010) provide vector autoregression evidence using postwar U.S. data, which suggests that a news shock much like the one considered here drives inflation down. We view this as support for our news-shock interpretation of a stock market boom.

6For example, according to one biography, “from the beginning, Bezos sought to increase market share as quickly as possible, at the expense of profits” (see http://www.achievement.org/autodoc/page/bez0bio-1).

7Recent work has been stimulated by the papers of Beaudry and Portier (2004, 2006).

8 Valerie Ramey (2009, forthcoming) has done important recent work fleshing out the idea that government spending shocks are heralded by early signals.

9 See Michelle Alexopoulos (2007) for a discussion of how the publishing indus-try broadcasts signals about future technology changes.

10The intuition for this finding is simple. The natural rate of interest corre-sponds roughly to the expected growth rate of consumption from the present to the future. In New Keynesian models fit to the data, one estimates that shocks are highly persistent. For example, an unexpected jump in technology today creates the expectation of a roughly equal jump in technology in the future. The absence of substantial internal persistence in models then implies a similar pattern for con-sumption. This means that the expected growth rate of consumption—and, hence, the natural rate—is relatively insensitive to the shocks incorporated in standard econometric analyses. But note how very different a shock to expectations about


the future is. Nothing happens in the present. Something happens only in the future. Such a shock has the potential to have a big impact on the intertemporal slope of consumption.

11Here, we are implicitly assuming that price frictions play a more important role than wage frictions. If there were only wage frictions, then optimal monetary policy would simply not reduce the interest rate. We elaborate on these observa-tions in the following subsection.

12Again, we implicitly assume that the key frictions are price frictions. The case for price stabilization weakens if there are significant wage frictions. With only wage frictions, it is desirable to stabilize wages, not prices (Erceg, Henderson and Levin, 2000).

13This is a theme also developed by Shirakawa (2010), governor of the Bank of Japan.

14Schwert’s (1990) annual index of common stock prices is available for the period 1802-1999 as series Cj797 in the Historical Statistics of the United States, Mille-nium Online Edition. The consumer price index is series Cc1 in the same source.

15These are series Ca9 in Historical Statistics of the United States, Millennial Edition Online. We also considered the annual industrial production index con-structed in Davis (2004), which spans the period 1790 to 1915. Using this variable instead of gross domestic product (GDP) has very little impact on the results. The mean logarithmic growth rate of industrial production in the non-boom, non-Civil War part of the period 1802-1914 is 4.0 percent. The corresponding mean growth rate in each of our nine stock market boom periods is 1.2, 7.2, 6.8, 5.3, 11.8, 4.3, 8.5, 4.9 and -0.2 percent, respectively. These results are very similar to those reported for GDP in Table 2.

16Our measure of credit splices together three time series. For the years 1819-1834, we add series Cj148 (loans and discounts, state banks) and Cj189 (loans and discounts, Second Bank of the United States) from Historical Statistics of the United States, Millennial Edition Online. For the period 1834 to 1896, we use “total loans of all banks,” series X582 in Chapter X (“Banking”), page 1019 of Bicentennial Edition: Historical Statistics of the United States, Colonial Times to 1970, part 2 (HSUS). For the period 1896 to 1914, we use series X582 in the table on page 1020 in HSUS. Though the latter two series have the same name, the coverage of the first series is incomplete, by comparison with that of the second series (see page 1011 of HSUS for details). To explain how we spliced the data, let xt denote the first data series, yt the second and zt the third. Let t = t1 and t = t2 denote the (unique) date when the first two and the second two series overlap, respectively. We let a = yt1 /xt1

and set xt = axt. We let b = zt2 /yt2

and setˆ , ˆ .x bx y byt t t t" "� Our data series is then, ( ˆ , ˆ ,x y zt t t

): Our measure of cred-it differs from the one used in Bordo and Wheelock (2004) (see http://research.stlouisfed.org/publications/review/04/11/0411dwd.xls). First, in constructing yt, they


compute the sum of series X582 (i.e., the only series we use) plus series X583 (“to-tal investments of all banks”). We did not include series X583 because, according to page 1011 in HSUS, X583 is composed primarily of government debt, while we seek a measure of nonfinancial business borrowing. In any case, our results are not sensitive to the inclusion of X583. We also differ from Bordo and Wheelock in that we use data from before 1834.

17We identified these as follows. Using Google in Windows Internet Explorer (32 bit or 64 bit versions), we typed “panic of 18” and Google completed the phrase with 10 panics. To select the episodes in the pre-World War I portion of the 20th century, we performed the same Windows Explorer exercise. We used all the panics identified in this way except the panic of 1901 which was too small to show up as a drop in the stock market in our annual data set.

18Our data on the CPI and the real value of the S&P composite price index were taken from Robert Shiller’s Web page. Data on pre-World War II U.S. real, quar-terly, seasonally adjusted GNP were taken from the online data appendix to Bordo and Wheelock (2004). The latter data were spliced with analogous GNP data for the post-World War II period taken from the Federal Reserve Bank of St. Louis’ online database. For the period after 1946, we measured credit with the flow of funds data, “credit market instruments,” taken from Haver Analytics (mnemonic, AL14TCR5). For the period, 1946-1951, these data are stocks pertaining to the fourth quarter of each year. We used log-linear interpolation to estimate observa-tions for the first, second and third quarters in the period, 1946-1951. For the pe-riod before 1945, our credit market data are observations on corporate debt, which corresponds to variable cj876, taken from Historical Statistics of the United States, Millennial Edition Online. The pre-1945 data were log-linearly interpolated and spliced with the post-1945 data. The credit data were converted into real terms by dividing by the CPI.

19Both series were obtained from the International Monetary Fund’s Internation-al Financial Statistics data set. The share prices correspond to series code 15862...ZF... and the CPI corresponds to series code 15864...ZF..... The real share prices displayed in Chart 3 have been converted to real terms by dividing by the CPI.

20The call money rate was obtained from International Financial Statistics The International Monetary Fund’s gap data were found at http://www.imf.org/external /pubs/ft/weo/2010/01/weodata/weoselser.aspx?c=158&t=1.

21The results are qualitatively similar for a range of values of the coefficients on inflation and the gap, and the smoothing parameter.

22The empirical model of the next section includes news about periods further into the future than just one period. We work with one-period-ahead news in this section because our objective is to keep things sufficiently simple that the basic ideas are apparent.


23It can be shown that the “efficient” equilibrium is the Ramsey optimal equilib-rium, in the case that there is no initial price dispersion. The Ramsey equilibrium considers just the private sector optimality and market clearing conditions, and leaves out a specification of the monetary policy rule. In addition, there is a subsidy on the employment of labor, designed to address the distortions associated with monopoly power. The endogenous variables, including the tax subsidy, are now underdetermined, as there are two more variables than equations. The Ramsey optimal equilibrium is the configuration of variables that satisfies the private sector equilibrium conditions and maximizes social welfare.

24The law of motion for at implies:

Eta t+j=p ja t+W�j -1 ]t1 .

25For an extensive discussion of the relationship between the time series represen-tation of at and the sign of the contemporaneous inflation and output response to

]t0, see Christiano, Trabandt and Walentin (forthcoming).

26Let ct denote consumption, scaled by exp (at ), and let Ht denote hours worked.

The Appendix shows that ˆ ˆ ˆ .c H xt t t" " Then,

1+( ) = +σ σL t t L tx c Hˆ ˆ ˆ ,

which is the log-linear expansion of the (scaled) marginal rate of substitution be-tween consumption and leisure when utility is logarithmic in consumption, and constant elasticity in labor.

27These results can be verified by considering the usual static monopoly diagram with price on the vertical axis and quantity on the horizontal, depicting demand, marginal revenue and marginal cost. Then, examine the effects of a given upward shift in marginal cost under two scenarios: one in which the marginal cost curve is flat and another in which the marginal cost curve is steep.

28The impact of increasing marginal costs on the slope of the price Phillips curve has received a lot of attention in the literature on firm-specific capital. To our knowl-edge, the effect was first noted in Sbordone (2002), and also discussed in Altig, Chris-tiano, Eichenbaum and Linde (2005), and de Walque, Smets, and Wouters (2005).

29The model is a version of the one in Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2007).

30There is a second equilibrium condition that the price of capital must satisfy, in addition to the present discounted value relation. The second condition is the requirement that in competitive markets, the price of capital must equal its mar-ginal cost (i.e., the “Tobin’s q” relation). The signal shock creates the expectation that technology will be high in the future, and that investment will be strong in


response. Given the adjustment cost specification, there is a gain to having in-creased investment in advance. This gain manifests itself in the form of a reduc-tion in the current marginal cost of producing capital. Given our assumption that capital is traded in competitive markets, the reduction in cost is passed on in the form of a reduction in price.

31 The object, St. It is the multiplier on the household budget constraint, (A.6), in the Lagrangian representation of its problem.


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Monetary Policy and Stock Market Boomslchrist/research/...Monetary Policy and Stock Market Booms 87 According to the conventional wisdom, it is only the “excessive” booms that